Demonstration of analyzers for multimode photonic time-bin qubits

We demonstrate two approaches for unbalanced interferometers as time-bin qubit analyzers for quantum communication, robust against mode distortions and polarization effects as expected from free-space quantum communication systems including wavefront deformations, path fluctuations, pointing errors, and optical elements. Despite strong spatial and temporal distortions of the optical mode of a time-bin qubit, entangled with a separate polarization qubit, we verify entanglement using the Negative Partial Transpose, with the measured visibility of up to $0.85\pm 0.01$. The robustness of the analyzers is further demonstrated for various angles of incidence up to $0.2^{\circ }$. The output of the interferometers is coupled into multimode fiber yielding a high system throughput of 0.74. Therefore, these analyzers are suitable and efficient for quantum communication over multimode optical channels.

Figure 1

Time-bin-based quantum communication in a turbulent free-space channel. When a time-bin-encoded photon, whose path is deviated by atmospheric turbulence as well as telescopes misalignment, enters a time-bin receiver with variable angle of incidence $\alpha$, a lateral offset $\delta \left(\alpha \right)$ occurs between the paths (red and blue line) at the interferometer exit. This is due to the receiver length asymmetry and reduces the interference quality, resulting in lower distinguishability of the time-bin states in superposition bases. Turbulence-induced spatial-mode distortions further lower the interference visibility (see text for details).

Figure 2

Our multimode time-bin qubit analyzers (MM-TQA). (a) Image of the incident single-mode Gaussian beam, captured with a beam-profiling camera (WinCamD-UCD12). (b) Image of the incident multimode beam, generated by a 1 m-long step-index multimode fiber (Thorlabs M43L01). (c) Schematic diagram of our MM-TQA using imaging optics. Temporal separation between paths is set to 2.0 ns ($\mathrm{\Delta }{l}_0=0.60\mathrm{m}$). $\mathrm{f}$ denotes a focal length of the lens. (d) Interference visibility for the single-mode beam with (green circles) and without (black squares) relay optics. Measured visibilities are in good agreement with theoretical prediction of Eq. (2) (dashed black line). (e) Interference visibility for the multimode beam with (green circles) and without (black squares) relay optics. (f) Schematic diagram of our MM-TQA with different refractive indices for each path. Temporal separation between paths is set to 0.57 ns ($\mathrm{\Delta }{l}_0=0.17\mathrm{m}$). (g) Interference visibility for the single-mode beam with (green circles) and without (black squares) glass. Measured visibilities are in good agreement with theoretical prediction of Eq. (2) (dashed black line). (h) Interference visibility for the multimode beam with (green circles) and without (black squares) glass. The uncorrected TQA visibilities in (g) and (h) are higher than in (d) and (e), because shorter path-length difference introduces less path distinguishability. Uncertainties are smaller than symbol size.

Figure 3

Experimental setup. The polarization-entangled photon-pair source (EPS) is described in the text. For projection measurements, photon A is directed to a polarization-qubit analyzer (PQA), consisting of a quarter-wave plate (QWP), a half-wave plate (HWP), a polarizing beam splitter (PBS), and silicon avalanche photodiodes (Si-APDs). After reflection at a dichroic mirror (DM), via a flip mirror (FM), photon B is sent either to a PQA or a time-bin qubit converter (${\mathrm{TQC}}_{1\left(2\right)}$) followed by a multimode fiber (MMF) and a multimode time-bin qubit analyzer (MM-${\mathrm{TQA}}_{1\left(2\right)}$). All detection signals are sent to a time tagger for data analysis.

Figure 4

Experimental results for entangled photons analyzed with our MM-TQAs. (a) Spatial mode of photon A before entering the MM-TQA. The image is captured with an electron multiplier CCD camera (Hamamatsu C9100-13). (b) Joint detections for the projection $+{\mathrm{Z}}_{\mathrm{A}}\otimes \pm {\mathrm{Z}}_{\mathrm{B}}$ (dotted green lines) and $-{\mathrm{Z}}_{\mathrm{A}}\otimes \pm {\mathrm{Z}}_{\mathrm{B}}$ (solid orange lines) as a function of detection-time difference between the photon A and B. (c) Joint detections for the projection $+{\varphi }_{{\mathrm{A}}^{\prime }}\otimes {\varphi }_{\mathrm{B}}$ (green squares) and $-{\varphi }_{{\mathrm{A}}^{\prime }}\otimes {\varphi }_{\mathrm{B}}$ (orange circles) as a function of phase ${\varphi }_{{\mathrm{A}}^{\prime }}$ of a polarization qubit using motorized wave plates. Visibilities ${\mathcal{V}}_{\pm \varphi }$ are obtained from sinusoidal fittings. Single counts remain essentially constant as we scan the phase. (d) The measurements (b) and (c) are repeated for different AOIs. Red circles and blue squares are average visibilities obtained with Method 1 and 2, respectively. Solid red and dotted blue lines are reference visibilities measured directly with our source of polarization-polarization entanglement prior to the measurement of time-polarization entanglement using Method 1 and 2, respectively. We maintain high entanglement visibility (close to source visibility) despite the high multimode nature of incoming photons.

Figure 5

Phase stability of our MM-TQA, Correlation visibilities ${\mathcal{V}}_{{\varphi }_{{\mathrm{A}}^{\prime }},{\varphi }_{\mathrm{B}}}$ (green circles) are measured using Method 1 as the AOI is continuously varied from $-0.2^{\circ }$ to $+0.2^{\circ }$ over 20 sec. The inset shows the calculated visibilities without relay optics as a function of AOI. Due to AOI-induced phase fluctuations, the value rapidly changes with AOI and yields an average value of zero (red circle). These phase fluctuations are corrected with relay optics, allowing a near constant visibility.

Figure 6

Entanglement verification. The Negative Partial Transpose (NPT) criterion [28, 29] is used to obtain the required entanglement visibilities, certifying the presence of entanglement in an arbitrary 2 $\times$ 3-dimensional quantum state. Our experimental results for various angles with Method 1 (red circles) and Method 2 (blue squares) are well above the classical bound (black line).

Figure 7

Estimation of the CHSH-Bell parameter. (a) Long-dashed yellow and short-dashed light blue lines are coincidences from joint projections $(\mathrm{Z}+\mathrm{X})\otimes +\mathrm{Z}$ (early temporal bin), long dash-dotted green and short dash-dotted orange lines from $(\mathrm{Z}+\mathrm{X})\otimes -\mathrm{Z}$ (late temporal bin), and solid purple and dotted blue lines from $(\mathrm{Z}+\mathrm{X})\otimes \varphi$ (middle temporal bin). Dashed black lines are times at which maximal expectation values are extracted. (b) Surface plot of calculated expectation values for the projections in (a) between any two points in time. (c) Long-dashed yellow and short-dashed light blue lines are coincidences from joint projections $(\mathrm{Z}-\mathrm{X})\otimes +\mathrm{Z}$, long dash-dotted green and short dash-dotted orange lines from $(\mathrm{Z}-\mathrm{X})\otimes -\mathrm{Z}$, and solid purple and dotted blue lines from $(\mathrm{Z}-\mathrm{X})\otimes \varphi$. Dashed black lines are times at which maximal expectation values are extracted. (d) Surface plot of calculated expectation values for the projection measurements in (c). The Bell-CHSH parameter is calculated using the maximum expectation values. The measurement duration is chosen arbitrarily and yields a violation of the inequality of 2.42 $\pm$ 0.05.

Figure 8

Long-term phase stability of our MM-TQA. Red squares are expectation values for projection measurements ${\varphi }_{{\mathrm{A}}^{\prime }}\otimes {\varphi }_{\mathrm{B}}$ ($V_{{\varphi }_{A^{\prime }},{\varphi }_B}$) and blue hexagons for ${\varphi }_{{\mathrm{A}}^{\prime }}+\pi /2\otimes {\varphi }_{\mathrm{B}}$ ($V_{{\varphi }_{A^{\prime }}+\pi /2,{\varphi }_B}$). Each measurement is averaged over 3 min. Black circles are average correlation visibility values $\sqrt{V_{{\varphi }_{A^{\prime }},{\varphi }_B}^2+V_{{\varphi }_{A^{\prime }}+\pi /2,{\varphi }_B}^2\phantom{{}^l}}$. The phase drifts slowly, on the order of $\pi$/2 over an half an hour, showing the stability of our MM-TQA. The average expectation value is always higher than the required expectation value for entanglement verification.